Why do you think that "the proof of the transcendence should also work with the famous Thue-Siegel-Roth theorem"? That is, why do you think that this number is very well approximable? Most numbers are not very well approximable and transcendental. You should think of the Thue-Siegel-Roth theorem as a result about algebraic numbers, not about transcendental ones.
–
GH from MOAug 21 '12 at 12:53

2 Answers
2

There are applications of Thue--Siegel--Roth and related results to this question (but it seems that they use in some form or another the rational approximations already present in Mahler's work).

For example, Baker in "On Mahler's classification of transcendental numbers." Acta Math. 111 1964 97–120,
shows that this constant is not a U-number using a result he describes as "This extends a theorem of LeVeque [..] which itself is a generalisation of Roth's Theorem"

However, then for the proof of this application it reads "It is clear from the proof of these results that the hypotheses of Theorem 1 are satisfied..." Where 'the proof of these results' refer to Mahler's.

And much more recently, Adamczewski and Bugeaud in "Mesures de transcendance et aspects quantitatifs de la méthode de Thue-Siegel-Roth-Schmidt." Proc. Lond. Math. Soc. 101 (2010), no. 1, 1–26 generalise this result of Baker together with a classical result of Ridout to then show that the generalizations of the Champernonwne Constant (any base, any suitable polynomial) are all S or T numbers.
See section 3 of the paper, in particular Théorème 3.2. But again the proof uses the rational approximations of the original transcendence proofs (by Mahler); cf. the final paragraph of Section 3.

In a more unexpected way, Mahler's
arguments led to the following amusing
result: Suppose $f$ is a non-constant
polynomial taking integer values at
the nonnegative integers. Then the
concatenated decimal $$
> \phi=0.f(1)f(2)f(3)\dots $$ is
transcendental. In particular
Champernowne's normal number $$
> 0.123\dots910111213\dots $$ is transcendental. Mahler's argument
relies on the observation that one
readily obtains rational
approximations to $\phi$ with
denominators high powers of the base
10, thus composed of the primes 2 and
5 alone. Perhaps disappointingly,
Roth's definitive form of the
Thue--Siegel inequalities permits a
more immediate argument obviating the
need for an appeal to the $p$-adic
results.

This is to say that Roth's argument is more superior than Mahler's but it appeared some 20 years later...